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310 Direct-Coupled Filters
Table 8.4.
Maximally Flat QL Values
N QLI Q" Qu QL' Q" QLo 01. Product FpQLNf- I / N
Lossless Source
2 2. 1.000 2.00 0.7071
3 3. 2.667 1.000 8.00 0.500
4 4. 4.121 2.828 1.000 46.63 0.3827
5 5. 5.483 4.472 2.894 1.000 354.90 0.3090
6 6. 6.797 6.000 4.643 2.928 1.000 3327.00 0.2588
Lossy Source
2 I. 1.000 1.000 1.4140
3 I. 2.000 1.000 2.000 1.0000
4 I. 2.414 2.414 1.000 5.8280 0.7654
5 I. 2.618 3.236 2.618 1.000 22.1800 0.6180
6 I. 2.732 3.732 3.732 2.732 1.000 104.0000 0.5176
The location fro +jim of the rnth zero is
r m =Fp,-'/Nsin(2m-I)8, (8.75)
im=Fp,-I/Ncos(2m-I)8. (8.76)
In these equations, m= 1,2, ... ,N/2 when N is even, and m= 1,2, ... ,
(N + 1)/2 when N is odd. The recursive relationship for normalized loaded-Q
values is given in Appendix G.
Unlike the Chebyshev or undercoupled cases, the Bu tterworth passband
constant QLNFp comes from a single, simple expression. The B'!tterworth Q
distributions and constants for values of N from 2 to 6 are given in Table 8.4.
8.4.3. The Fano Undercoupled Response Shape. The Fano response shape is
defined by
where
and
k= q
sinh(0.8814N)
(8.77)
(8.78)
q=(IO L ,/IO_I)'/2. (8.79)
Normalizing bandwidth F d
in (8.77) corresponds to loss L d , the dB droop. It
has an upper bound:
L d " 10 10glO[ I + sinh 2 (0.8814N) J. (8.80)